At the start of his summation, Atticus's tone shifts from impersonal, objective lawyer to a more personal tone.
“Gentlemen,” he said. Jem and I again looked at each other: Atticus might have said, “Scout.” His voice had lost its aridity, its detachment, and he was talking to the jury as if they were folks on the post office corner.
With the impersonal, objective tone, Atticus wanted to present the evidence. With this more personal tone, he wants to appeal to the humanity of the jury in the hopes that they would ignore their racial bias and deliver a just verdict.
Having grown up in a society where/when we know racism is repugnant, I would have no problem recognizing Tom's innocence. Hypothetically speaking, if I had grown up in a southern town like Maycomb during the 1930s, indoctrinated with segregation and racism, I like to think that I would rise above any socially cultivated racism, especially in a courtroom, and defend Tom's innocence. In either case, rising above it or succumbing to it, I think Atticus would convince me of Tom's innocence. He made a sound argument throughout the trial, an argument more logical and consistent than Mr. Gilmer's.
It is reasonable to suppose that some, most, or all of the jury members supposed or were convinced of Tom's innocence. In this sense, it wasn't that Atticus's words were not convincing. It was simply that the jury was unwilling to rise above that endemic racism.
Atticus reiterates the evidence and also implores the jury to recall the inconsistent testimonies of Mayella and Bob Ewell as well as the calm and logical testimony by Tom. Atticus ends by putting the onus on the jury that it is their duty to make a just decision:
I am confident that you gentlemen will review without passion the evidence you have heard, come to a decision, and restore this defendant to his family. In the name of God, do your duty.
Atticus had two goals with this case. The first was to make a convincing argument that Tom was innocent. I think he succeeded in doing so and I believe that, given the hypothetical that I was born in the early 1900s in the south, I would also be convinced of Tom's innocence as a member of that jury during that time period. That is the direct answer to your question. But Atticus had a second goal and that was to implore the jury to "review without passion" the evidence of the trial. That is, to review without racial bias. He may have convinced the jury of Tom's innocence but even if he did, he was not able to convince them to ignore any racial bias they might have had.